Verifying the Implicitization Formulae for Degree N Rational Bézier Curves

This is a continuation of short communication$^{[1]}$. In [1] a verification of the implicitization equation for degree two rational Bézier curves is presented which does not require the use of resultants. This paper presents these verifications in the general cases, i.e., for degree $n$ rational Bézier curves. Thus some interesting interplay between the structure of the $n×n$ implicitization matrix and the de Casteljau algorithm is revealed.     

ON THE IRREDUCIBILITY OF THE DETERMINANT OF THE MATRIX OF MOVING PLANES

ABSTRACT

We show that, for generic bihomogeneous polynomials, the determinant of the matrix of moving planes is irreducible.     

VERIFYING THE IMPLICITIZATION FOR MULAE FORDEGREE n RATIONAL BEZIER CURVES

1.IntroductionInordertoinvestigateimplicitrepresentationsofparametriccurvesandsurfaces,thetraditionalalgebraicgeometrytheoryisalwaysused.Recentlysomeresearchreports,e.g.[1],showthattherisingBlossomingprinciple[213]ismoreintuitiveandefficientthanthemethodofalgebraicgeometryfortheimplicitization.Thispaperisacotinuationof[1].GivenadegreenplanerationalBenercurvewherePi~(wixi,fiji,fi)arethehomogeneousB6ziercontrolpoints.DenotebywhereP~(x,y,w)isthehomogeneouspoint.LetUsingpolynomialresultants,th…     

The implicitization problem for

We develop in this paper methods for studying the implicitization problem for a rational map defining a hypersurface in , based on computing the determinant of a graded strand of a Koszul complex. We show that the classical study of Macaulay resultants and Koszul complexes coincides, in this case, with the approach of approximation complexes and we study and give a geometric interpretation for the acyclicity conditions.Under suitable hypotheses, these techniques enable us to obtain the implicit equation, up to a power, and up to some extra factor. We give algebraic and geometric conditions for determining when the computed equation defines the scheme theoretic image of , and, what are the extra varieties that appear. We also give some applications to the problem of computing sparse discriminants.